PlanetPhysics/Legendre Polynomials

The Legendre polynomials generate the [[../Power/|power]] series that solves Legendre's [[../DifferentialEquations/|differential equation]]:

$(1-x^2)P^{″}(x)-2x{P}^{\prime }(x)+n(n+1)P(x)=0.$

This [[../DifferentialEquations/|ordinary differential equation]] with variable coefficients is named in honor of Adrien-Marie Legendre (1752-1833). While quite literally following in the footsteps of Laplace, he developed the Legendre polynomials in a paper on celestial [[../Mechanics/|mechanics]]. In a strange tangled web of fate, the Legendre polynomials are heavily used in electrostatics to solve [[../FluorescenceCrossCorrelationSpectroscopy/|Laplace's equation]] in spherical coordinates

${\mathrm{\nabla }}^2{\mathrm{\Phi }}_{sph}=0$

The series can be easily generated using the Rodrigues' [[../Formula/|formula]] $P_n(x)=\frac{1}{2^{n}n!}\frac{d^n}{d{x}^n}(x^2-1{)}^n.$

The first six polynomials are:

$P_0(x)=1$\\ $P_1(x)=x$\\ $P_2(x)=\frac{1}{2}(3x^2-1)$\\ $P_3(x)=\frac{1}{2}(5x^3-3x)$\\ $P_4(x)=\frac{1}{8}(35x^4-30x^2+3)$\\ $P_5(x)=\frac{1}{8}(63x^5-70x^3+15x)$\\

Not yet done....

References

[1] Lebedev, N. "Special [[../Bijective/|functions]] \& Their Applications." Dover Publications, Inc., New York, 1972.

[2] Jackson, J. "Classical Electrodynamics." John Wiley \& Sons, Inc., New York, 1962.

\htmladdnormallink{http://www-groups.dcs.st-and.ac.uk/\~{}history/Biographies/Legendre.html}{http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Legendre.html} \htmladdnormallink{http://astrowww.phys.uvic.ca/\~{}tatum/celmechs.html}{http://astrowww.phys.uvic.ca/~tatum/celmechs.html} \htmladdnormallink{http://www.du.edu/\~{}jcalvert/math/legendre.htm}{http://www.du.edu/~jcalvert/math/legendre.htm} http://en.wikipedia.org/wiki/Legendre_polynomials